Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.59 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&3\\6&7\end{bmatrix}$, $\begin{bmatrix}13&17\\22&19\end{bmatrix}$, $\begin{bmatrix}17&4\\6&7\end{bmatrix}$, $\begin{bmatrix}23&16\\20&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 99x - 378 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{156x^{2}y^{6}+11284767x^{2}y^{4}z^{2}+51980096952x^{2}y^{2}z^{4}+43626516244137x^{2}z^{6}+10542xy^{6}z+247268592xy^{4}z^{3}+744027068241xy^{2}z^{5}+501062829774858xz^{7}+y^{8}+410896y^{6}z^{2}+3569279040y^{4}z^{4}+5087920043052y^{2}z^{6}+1435822394391657z^{8}}{12x^{2}y^{6}-1377x^{2}y^{4}z^{2}-17496x^{2}y^{2}z^{4}+59049x^{2}z^{6}-18xy^{6}z+7776xy^{4}z^{3}+111537xy^{2}z^{5}-354294xz^{7}+y^{8}-1296y^{6}z^{2}+93312y^{4}z^{4}+1023516y^{2}z^{6}-3720087z^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0.s.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bv.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.1.c.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.cz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.jk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.jy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ma.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.mg.1 | $24$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
| 24.96.5.es.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 48.48.2.ea.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.eb.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.ec.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.ed.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.ee.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.ef.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.eg.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.eh.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 120.48.1.biu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.biy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bka.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bke.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bto.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bts.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.buu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.buy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.sc.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.oda.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.faq.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bis.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.biw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bjy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bkc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.btm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.btq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bus.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.buw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.ma.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 240.48.2.ek.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.el.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.em.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.en.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.eo.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ep.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.eq.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.er.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.48.1.bis.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.biw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bjy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bkc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.btm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.btq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bus.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.buw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.ke.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.biu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.biy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bka.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bke.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bto.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bts.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.buu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.buy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |